Let $T$ be some random variable on $[0,1]$, and define \begin{equation} \alpha(t) \triangleq \mathbb{E}[T \vert T\le t],\\ \beta(t) \triangleq \mathbb{E}[T \vert T>t], ~t\in[0,1]. \end{equation} Consider the objective function \begin{equation} f(t)=F_{T}(t)\cdot h_b(\alpha(t)) +(1-F_{T}(t)) \cdot h_b(\beta(t)), \end{equation} where $F_{T}(t)=\mathbb{P}\left\{ T\le t\right\}$ is the CDF of $T$ and $h_b(x)=-x\log (x)-(1-x)\log(1-x)$ is the binary entropy function.
My question is how to prove that there exists a number $\zeta\in[0,1]$ such that the function $f(t)$ is non-increasing on $[0,\zeta]$ and non-decreasing on $[\zeta,1]$. Comprehensive numerical results have verified this argument.
